Pythagorean Theorem and it's Converse

Front

What is the Pythagorean Theorem?

Back

The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). It can be expressed as: c² = a² + b².

Front

What is the converse of the Pythagorean Theorem?

Back

The converse of the Pythagorean Theorem states that if in a triangle, the square of the length of one side is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle.

Front

If a right triangle has legs of lengths 3 cm and 4 cm, what is the length of the hypotenuse?

Back

5 cm (Using the Pythagorean Theorem: 3² + 4² = 9 + 16 = 25, so c = √25 = 5 cm).

Front

How do you find the length of a missing side in a right triangle?

Back

To find the length of a missing side, use the Pythagorean Theorem: c² = a² + b². Rearrange the formula to solve for the missing side.

Front

What is the distance formula derived from the Pythagorean Theorem?

Back

The distance formula is used to find the distance between two points (x₁, y₁) and (x₂, y₂) in a coordinate plane: d = √((x₂ - x₁)² + (y₂ - y₁)²).

Front

In a right triangle, if one leg is 6 cm and the hypotenuse is 10 cm, what is the length of the other leg?

Back

8 cm (Using the Pythagorean Theorem: 10² = 6² + b², so 100 = 36 + b², thus b² = 64, and b = √64 = 8 cm).

Front

What is the relationship between the sides of a 30-60-90 triangle?

Back

In a 30-60-90 triangle, the lengths of the sides are in the ratio 1:√3:2. The shortest side is opposite the 30° angle, the longer leg is opposite the 60° angle, and the hypotenuse is opposite the right angle.

Front

What is the relationship between the sides of a 45-45-90 triangle?

Back

In a 45-45-90 triangle, the lengths of the legs are equal, and the hypotenuse is √2 times the length of each leg.

Front

How can the Pythagorean Theorem be applied in real life?

Back

The Pythagorean Theorem can be used in various real-life situations, such as determining the height of a tree using its shadow, calculating the distance between two points, or finding the length of a ladder needed to reach a certain height.

Front

If a ladder is placed 10 feet away from a wall and reaches a height of 24 feet, how long is the ladder?

Back

26 feet (Using the Pythagorean Theorem: 10² + 24² = c², so 100 + 576 = c², thus c = √676 = 26 feet).